Topic 2.6 Gravitational Force: use Newton's law of universal gravitation to find the force between masses, and relate this to weight and the gravitational field strength near a planet's surface.

What this topic is asking

The College Board (Topic 2.6) wants you to use Newton's law of universal gravitation to find the attractive force between two masses, to understand its inverse-square dependence on distance, and to relate gravity to weight and the gravitational field strength $g$ near a planet. A central exam theme is the difference between mass (which never changes) and weight (which depends on where you are).

Newton's law of universal gravitation

Gravity is always attractive and always acts between every pair of masses, though it is only noticeable when at least one mass is astronomically large. The forces on the two masses form a Newton's-third-law pair: equal in magnitude, opposite in direction, regardless of how different the masses are.

The inverse-square law

The inverse-square behavior is why gravity weakens rapidly with altitude in astronomical terms but is nearly constant over the few kilometers near a planet's surface (the change in $r$ is tiny compared with the planet's radius). This is what lets us treat $g$ as constant in everyday projectile and free-fall problems.

Gravitational field strength and weight

The gravitational field strength $g$ at a location is the gravitational force per unit mass placed there:

where $M$ is the mass of the planet (or star) and $r$ the distance from its center. Near Earth's surface, $g \approx 9.8$ N/kg. The weight of an object is then the gravitational force on it:

Notice that $g$ here is the same quantity as the free-fall acceleration $9.8$ m/s squared from kinematics: an object in free fall has only its weight acting on it, so $F_{net} = mg = ma$ gives $a = g$. The units N/kg and m/s squared are equivalent.

Why mass and weight must be kept apart

The exam returns again and again to the distinction between mass and weight, because the two are easy to conflate in everyday speech but physically different. Mass is the amount of matter, a measure of inertia, fixed in kilograms no matter where the object is; it is the $m$ in $F = ma$. Weight is the gravitational force on that mass, $W = mg$, measured in newtons, and it changes with the local field strength. An astronaut who weighs about $590$ N on Earth weighs only about $98$ N on the Moon (where $g \approx 1.6$ N/kg), yet has exactly the same $60$ kg mass in both places, and so the same resistance to being pushed. "Weightlessness" in orbit is not the absence of gravity (gravity is what holds the orbit) but free fall, in which there is no supporting normal force to give the sensation of weight. Keeping mass as the inertia term and weight as a force lets you put weight correctly on a free-body diagram (always $mg$ downward toward the planet's center) and reserve mass for the $F_{net} = ma$ step.

Try this

Q1. Calculate the weight of a $75$ kg person on Earth ($g = 9.8$ N/kg). [2 points]

• Cue. $W = mg = (75)(9.8) = 735$ N downward.

Q2. State what happens to the gravitational force between two masses if one mass is tripled (distance unchanged). [1 point]

• Cue. The force triples, since $F$ is proportional to each mass.